arXiv subjects

Salvador Addas-Zanata

Publications and source records attributed to Salvador Addas-Zanata.

11 recordsLinked to original sources

arXiv2025

On $C^r$-generic twist maps of ${\rm T^2}$

We consider twist diffeomorphisms of the torus, $f:{\rm T^2\rightarrow T^2,}$ and their vertical rotation intervals $ρ_V(\widehat{f})=[ρ_V^{-},ρ_V^{+}],$ where $\widehat{f}$ is a lift of $f$ to the vertical annulus or cylinder. We show that $C^r$-generically for any $r\geq 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when $f$ is area-preserving, $C^r$-generically $ρ_V^{-}<ρ_V^{+}.$ Also, for any twist map $f$, $\widehat{f}$ a lift of $f$ to the cylinder, if $ρ_V^{-}<ρ_V^{+}=p/q$, then there are two possibilities: either $\widehat{f}^q(\bullet)-(0,p)$ maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the Curve Intersection Property. In the first case, $ρ_V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $ρ_V^{+}(\widehat{f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $ρ_V^{-}<ρ_V^{+}=p/q,$ we present some consequences of the existence of the free loop for $\widehat{f}^q(\bullet)-(0,p)$, related to the description and shape of the attractor-reppeler pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated.

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arXiv2023

Mathers regions of instability for annulus diffeomorphisms

Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\widetilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^*$ whose boundary intersects both boundary components of of $A$, and points $z^+$ and $z^-$ in $A^*$, such that the positive (resp. negative) orbit of $z^+$ converges to a set contained in the upper (resp. lower) boundary component of $A^*$ and the positive (resp. negative) orbit of $z^-$ converges to a set contained in the lower (resp. upper) boundary component of $A^*$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms.

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arXiv2022

Homotopically unbounded disks for generic surface diffeomorphisms

In this paper we consider closed orientable surfaces $S$ of positive genus and $C^r$-diffeomorphisms $f:S\rightarrow S$ isotopic to the identity ($r\geq 1)$. The main objective is to study periodic open topological disks which are homotopically unbounded (i.e. which lift to unbounded connected sets in the universal covering). We show that these disks are not uncommon, and are related to important dynamical phenomena. We also study the dynamics on these disks under certain generic conditions. Our first main result implies that for the torus (or for arbitrary surfaces, with an additional condition) if the rotation set of a map has nonempty interior and is not locally constant, then the map is $C^r$-accumulated by diffeomorphisms exhibiting periodic homotopically unbounded disks. Our second result shows that $C^r$-generically, if the rotation set has nonempty interior (plus an additional hypothesis if the genus of $S$ is greater than $1$) a maximal periodic disk which is unbounded and has a rational prime ends rotation number must be the basin of some compact attractor or repeller contained in the disk. As a byproduct we obtain results describing certain periodic components of the complement of the closure of stable or unstable manifolds of a periodic orbit in the $C^r$-generic setting.

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arXiv2021

On Stable and Unstable Behaviours of Certain Rotation Segments

In this paper, we study non-wandering homeomorphisms of the two torus in the identity homotopy class, whose rotation sets are non-trivial line segments from $(0,0)$ to some totally irrational vector $(α,β)$. We show this rotation set is in fact a non-generic phenomenon for any $C^r$ diffeomorphisms, with $r \geq 1$. When such a rotation set does happen, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behavior. More precisely, the dynamics admits bounded deviation along the direction $-(α,β)$ in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small $C^0$-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism $f$ with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by rotation sets of nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavors interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.

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arXiv2018

A consequence of the growth of rotation sets for families of diffeomorphisms of the torus

In this paper we consider $C^\infty $-generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_t:\rm{T^2\rightarrow T^2}$ be such a family, $\widetilde{f}_t:\rm I\negthinspace R^2 \rightarrow \rm I\negthinspace R^2$ be a fixed family of lifts and $ρ(\widetilde{f}_t)$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval $I.$ We also assume that some rational point $(\frac pq,\frac rq)\in \partial ρ(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ and we want to understand consequences of the following hypothesis: For all $t>\overline{t},$ $t\in I,$ $(\frac pq,\frac rq)\in int(\partial ρ(\widetilde{f}_t)).$ Under these very natural assumptions, we prove that there exists a $f_{\overline{t}}^q$-fixed hyperbolic saddle $P_{\overline{t}}$ such that its rotation vector is $(\frac pq,\frac rq)$ and, there exists a sequence $t_i>\overline{t},$ $t_i\rightarrow \overline{t},$ such that if $P_t$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^u(\widetilde{P}_{t_i})$ (the unstable manifold) has quadratic tangencies with $W^s(\widetilde{P}_{t_i})+(c,d)$ (the stable manifold translated by $(c,d)),$ where $\widetilde{P}_{t_i}$ is any lift of $P_{t_i}$ to the plane, in other words, $\widetilde{P}_{t_i}$ is a fixed point for $(\widetilde{f}_{t_i})^q-(p,r),$ and $(c,d)\neq (0,0)$ are certain integer vectors such that $W^u(\widetilde{P}_{\overline{t}})$ do not intersect $W^s(\widetilde{P}_{\overline{t}})+(c,d).$ And these tangencies become transverse as $t$ increases.

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arXiv2018

A condition that implies full homotopical complexity of orbits

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ homotopic to the identity. A set of hypotheses is presented, called fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}),$ denoted $\widetilde{f},$ has complicated and rich dynamics. In this context we generalize results that hold for homeomorphisms of the torus homotopic to the identity when their rotation sets contain zero in the interior. In particular, we prove that if $f$ is a $C^{1+ε}$ diffeomorphism for some $ε>0$ and $π:\mathbb{D}\rightarrow S$ is the covering map, then there exists a contractible hyperbolic $f$-periodic saddle point $p\in S$ such that for any $\widetilde{p}\in π^{-1}(p),$ $$W^u(\widetilde{p}) \pitchfork W^s(g(\widetilde{p})) $$ for all deck transformations $g\in Deck(π).$ By $\pitchfork,$ we mean a topologically transverse intersection between the manifolds, see the precise definition in subsection 1.1. We also show that the homological rotation set of such a $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets, periodic orbits in the rational case, and $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. Something that implies, in case $f$ is area-preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

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arXiv2016

Rational Mode Locking for Homeomorphisms of the 2-Torus

Let $f:{\rm T^2\rightarrow T^2}$ be a homeomorphism homotopic to the identity, $\widetilde{f}:{\rm I}\negthinspace {\rm R^2\rightarrow I} \negthinspace {\rm R^2}$ be a fixed lift and $ρ(\widetilde{f})$ be its rotation set, which we assume to have interior. We also assume that some rational point $(\frac pq,\frac rq)\in \partial ρ(\widetilde{f})$ and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms, which are arbitrarily small $C^0$-perturbations of $f,$ denoted $f_1$ and $f_2,$ in a way that $(\frac pq,\frac rq)$ does not belong to the rotation set of $f_1$ and $(\frac pq,\frac rq)$ is contained in the interior of the rotation set of $f_2.$ We give two examples in this direction. The first is a $C^\infty $-diffeomorphism $f_{dissip},$ such that $(0,0)\in \partial ρ(\widetilde{f}_{dissip}),$ $f_{dissip}$ has only one fixed point with zero rotation vector and there are maps $f_1$ and $f_2$ satisfying the conditions above. The second is an area preserving version of the above, but in this conservative setting we obtain only a $C^0$ example. We also present two theorems in the opposite direction. The first says that if $f$ is area preserving and analytic, then there can not be $f_1$ and $f_2$ as above. The second result, implies that for a generic (in the sense of Brunovsky) one parameter family $% f_t:{\rm T^2\rightarrow T^2}$ of $C^1$-diffeomorphisms such that for some parameter $\overline{t},$ $ρ(\widetilde{f}_{\overline{t}})$ has interior, $(\frac pq,\frac rq) \in \partial ρ(\widetilde{f}_{\overline{t}})$ and $(\frac pq,\frac rq)\notin ρ(\widetilde{f}_t)$ for $t<\overline{t},$ then for all $t> \overline{t}$ sufficiently close to $\overline{t},$ $(\frac pq,\frac rq)\notin int(ρ( \widetilde{f}_{\overline{t}})).$

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arXiv2014

Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

In this paper we consider $C^{1+ε}$ area-preserving diffeomorphisms of the torus $f,$ either homotopic to the identity or to Dehn twists. We suppose that $f$ has a lift $\widetilde{f}$ to the plane such that its rotation set has interior and prove, among other things that if zero is an interior point of the rotation set, then there exists a hyperbolic $\widetilde{f}$-periodic point $\widetilde{Q}$$\in {\rm I}\negthinspace {\rm R^2}$ such that $W^u(\widetilde{Q})$ intersects $W^s(\widetilde{Q}+(a,b))$ for all integers $(a,b)$, which implies that $\bar{W^u(\widetilde{Q})}$ is invariant under integer translations. Moreover, $\bar{W^u(\widetilde{Q})}=\bar{W^s(\widetilde{Q})}$ and $\widetilde{f}$ restricted to $\bar{W^u(\widetilde{Q})}$ is invariant and topologically mixing. Each connected component of the complement of $\bar{W^u(\widetilde{Q})}$ is a disk with uniformly bounded diameter. If $f$ is transitive, then $\bar{W^u(\widetilde{Q})}=$${\rm I}\negthinspace {\rm R^2}$ and $\widetilde{f}$ is topologically mixing in the whole plane.

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arXiv2014

Persistence of fixed points under rigid perturbations of maps

Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic annulus diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$ and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$. Let $\tilde{f}_ε$ be the perturbation of $\tilde{f}$ by the rigid horizontal translation $(x,y)\mapsto (x+ε,y)$. We show that for all $ε>0$ sufficiently small we have ${\rm Fix} (\tilde{f}_ε)=\emptyset $. The proof follows from Kerékjártó's construction of Brouwer lines for orientation preserving homeomorphisms of the plane with no fixed points. This result turns out to be sharp with respect to the regularity assumption: there exists a diffeomorphism $f$ satisfying all the properties above, except that $f$ is not real-analytic but only smooth, so that the above conclusion is false. Such a map is constructed via generating functions.

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arXiv2014

Uniform bounds for diffeomorphisms of the torus and a conjecture of P. Boyland

We consider $C^{1+ε}$ diffeomorphisms of the torus, denoted $f,$ homotopic to the identity and whose rotation sets have interior. We give some uniform bounds on the displacement of points in the plane under iterates of a lift of $f,$ relative to vectors in the boundary of the rotation set and we use these estimates in order to prove that if such a diffeomorphism $f$ preserves area, then the rotation vector of the area measure is an interior point of the rotation set. This settles a strong version of a conjecture proposed by P. Boyland. We also present some new results on the realization of extremal points of the rotation set by compact $f$-invariant subsets of the torus.

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arXiv2012

Dynamics of homeomorphisms of the torus homotopic to Dehn twists

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

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